Asymptotic Analysis

We study asymptotic behavior of global positive solutions of the Cauchy problem for the semilinear parabolic equation u_t=Δu+u^p in R^N, where p>1+2/N, p(N-2)≤N+2. The initial data are of the form u(x, 0)=αϕ(x), where ϕ is a fixed function with suitable decay at |x|=∞ and α>0 is a parameter. There exists a threshold parameter α^* such that the solution exists globally if and only if α≤α^*. Our main results describe the asymptotic behavior of the solutions for α∈(0, α^*] and in particular exhibit the difference between the behavior of sub-threshold solutions (α<α^*) and the threshold solution (α=α^*).

This paper is devoted to the rigorous theory of nonlinear geometric optics for multiple shocks to a general N×N conservation law in one space variable. For the problem of multiple weak shocks perturbed by small amplitude, high-frequency oscillatory waves, we obtain that the leading profiles of oscillatory shocks are solution to an integro-differential system with free boundaries, the leading terms of shock fronts don't oscillate, and oscillations only appear in the leading terms shock speeds.

Our purpose is to investigate the effect of a thin dielectric coating on the scattering of a time-harmonic electromagnetic wave by a perfectly conducting metal. An asymptotic expansion of the solution of a model problem is performed relatively to the thickness of the dielectric shell. A suitable scaling as well as an adapted writing of the curl operator inside the thin shell make it possible to achieve an asymptotic expansion of the electromagnetic field at any order. Special stability estimates for Maxwell's equations relatively to the thickness of the dielectric shell then yield optimal error estimates outside the obstacle.

We consider an elliptic distributed quadratic optimal control problem with exact controllability constraints on a part of the domain which, in turn, is parametrized by a small parameter ε. The quadratic tracking type functional is defined on the remaining part of the domain. We thus consider a family of optimal control problems with state equality constraints. The purpose of this paper is to study the asymptotic limit of the optimal control problems as the parameter ε tends to zero. The analysis presented is in the spirit of the direct approach of the calculus of variations. This is achieved in the framework of relaxed problems. We finally apply the procedure to an optimal control problem on a perforated domain with holes of critical size. It is shown that a strange term in the terminology of Cioranescu and Murat (Prog. Nonlinear Diff. Eq. Appl., Vol. 31, Birkhäuser-Verlag, Boston, 1997, pp. 49–93) appears in the limiting homogenized problem.